We analyze a replicator-mutator model arising in the context of directed evolution [
Citation: |
Figure 2. (A): The first four approximations, for $ 0\leq t\leq 3 $, of the nonlocal term $ \overline{u}(t) $ computed via the fixed point iteration (28). (B): Numerical solution obtained by the method described in Section 5, starting from $ u_0 = \mathbb{1}_{[1/2,3/2]} $, with $ {\sigma ^2} = 1 $. The red points are the points on the graph $ u(t,\cdot) $ with abscissa $ x = \overline u(t) $. The green points are the maxima of $ u(t,\cdot) $. This reveals the dissymmetry of the solution
Figure 3. Vector field defined by the differential system (31) with $ {\sigma ^2} = 1 $, describing the dynamics of Gaussian solutions. In yellow, the set of initial conditions for which $ a $ blows up in finite time $ T^{\star} $ and in red, dark blue and light blue those for which both $ a $ and $ m $ are globally defined. The red dashed curve is the set of values defined by $ m_0 = -1/{(a_0\sqrt{2{\sigma ^2}})} $, for which $ a $ tends to infinity and $ m $ tends to zero as time goes to infinity. The dark blue region corresponds to the values leading to an anti-diffusion/diffusion behaviour. The light blue region corresponds to the values leading to a pure diffusion behaviour
[1] |
M. Alfaro and R. Carles, Explicit solutions for replicator-mutator equations: Extinction versus acceleration, SIAM J. Appl. Math., 74 (2014), 1919-1934.
doi: 10.1137/140979411.![]() ![]() ![]() |
[2] |
M. Alfaro and R. Carles, Replicator-mutator equations with quadratic fitness, Proc. Amer. Math. Soc., 145 (2017), 5315-5327.
doi: 10.1090/proc/13669.![]() ![]() ![]() |
[3] |
M. Alfaro and R. Carles, Superexponential growth or decay in the heat equation with a logarithmic nonlinearity, Dyn. Partial Differ. Equ., 14 (2017), 343-358.
doi: 10.4310/DPDE.2017.v14.n4.a2.![]() ![]() ![]() |
[4] |
M. Alfaro and M. Veruete, Evolutionary Branching Via Replicator–Mutator Equations, J. Dynam. Differential Equations, 2018.
![]() |
[5] |
F. H. Arnold, Design by Directed Evolution, Acc. Chem. Res, 1998.
![]() |
[6] |
I. Białynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Physics, 100 (1976), 62-93.
doi: 10.1016/0003-4916(76)90057-9.![]() ![]() ![]() |
[7] |
V. N. Biktashev, A simple mathematical model of gradual darwinian evolution: emergence of a gaussian trait distribution in adaptation along a fitness gradient, J. Math. Biol., 68 (2014), 1225-1248.
doi: 10.1007/s00285-013-0669-3.![]() ![]() ![]() |
[8] |
I. Bomze and R. Burger, Stability by mutation in evolutionary games, Games Econom. Behav., 11 (1995), 146-172.
doi: 10.1006/game.1995.1047.![]() ![]() ![]() |
[9] |
R. Bürger, On the maintenance of genetic variation: Global analysis of Kimura's continuum-of-alleles model, J. Math. Biol., 24 (1986), 341-351.
doi: 10.1007/BF00275642.![]() ![]() ![]() |
[10] |
R. Bürger, Mutation-selection balance and continuum-of-alleles models, Math. Biosci., 91 (1988), 67-83.
doi: 10.1016/0025-5564(88)90024-7.![]() ![]() ![]() |
[11] |
R. Bürger, Perturbations of positive semigroups and applications to population genetics, Math. Z., 197 (1988), 259-272.
doi: 10.1007/BF01215194.![]() ![]() ![]() |
[12] |
R. Carles and I. Gallagher, Universal dynamics for the defocusing logarithmic schrödinger equation, Duke Math. J., 167 (2018), 1761-1801.
doi: 10.1215/00127094-2018-0006.![]() ![]() ![]() |
[13] |
W. H. Fleming, Equilibrium distributions of continuous polygenic traits, SIAM J. Appl. Math., 36 (1979), 148-168.
doi: 10.1137/0136014.![]() ![]() ![]() |
[14] |
M.-E. Gil, F. Hamel, G. Martin and L. Roques, Mathematical properties of a class of integro-differential models from population genetics, SIAM J. Appl. Math., 77 (2017), 1536-1561.
doi: 10.1137/16M1108224.![]() ![]() ![]() |
[15] |
M.-E. Gil, F. Hamel, G. Martin and L. Roques, Dynamics of Fitness Distributions in the Presence of a Phenotypic Optimum: An Integro-differential Approach, HAL preprint, 2018.
![]() |
[16] |
K. Hadeler, Stable polymorphisms in a selection model with mutation, SIAM J. Appl. Math., 41 (1981), 1-7.
doi: 10.1137/0141001.![]() ![]() ![]() |
[17] |
J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems: Mathematical
Aspects of Selection, London Mathematical Society Student Texts, Cambridge University
Press, 1988.
![]() ![]() |
[18] |
M. Kimura, A stochastic model concerning the maintenance of genetic variability in quantitative characters, Proc. Natl. Acad. Sci., USA, 54 (1965), 731–736.
![]() |
[19] |
M. Nowak, N. Komarova and P. Niyogi, Evolution of universal grammar, Science, 291 (2001), 114-118.
doi: 10.1126/science.291.5501.114.![]() ![]() ![]() |
[20] |
K. Page and M. Nowak, Unifying evolutionary dynamics, J. Theoret. Biol., 219 (2002), 93-98.
doi: 10.1016/S0022-5193(02)93112-7.![]() ![]() ![]() |
[21] |
P. Schuster and K. Sigmund, Replicator dynamics, J. Theoret. Biol., 100 (1983), 533-538.
doi: 10.1016/0022-5193(83)90445-9.![]() ![]() ![]() |
[22] |
P. Stadler and P. Schuster, Mutation in autocatalytic reaction networks, J. Math. Biol., 30 (1992), 597-631.
doi: 10.1007/BF00948894.![]() ![]() ![]() |
[23] |
P. Taylor and L. Jonker, Evolutionary stable strategies and game dynamics, Math. Biosci., 40 (1978), 145-156.
doi: 10.1016/0025-5564(78)90077-9.![]() ![]() ![]() |
[24] |
A. Zadorin and Y. Rondelez, Natural selection in compartmentalized environment with reshuffling, arXiv preprint, arXiv: 1707.07461, 2017.
![]() |
Evolution of Gaussian solutions for
(A): The first four approximations, for
Vector field defined by the differential system (31) with
Case
Case
Case
Case